Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Wednesday, November 29, 2006

The Campers and the Bread

Marilyn vos Savant writes a weekly column for Parade Magazine instered into many US
papers. She's best known in
the math community for the popularization of the Monty Hall
Problem. Here is a question from her November 19th column (names added).

Carol get lost in the woods, where she ran into two campers, Alice and
Bob, who also were lost. Alice had three loaves of bread and Bob had
two loaves. They all agreed to share the bread equally. Carol was so
grateful that, when they found their way back to town, she gave the
campers $10,000 for saving her life. Alice said she should get $6,000
because she contributed 3/5 of the bread. Bob said that all had eaten
an equal amount so the campers should split the reward. To settle the
argument they visited the local wise man, a retired math
teacher. Which camper was right?

Think about the problem. Here is my short version of Marilyn's
answer.

Neither. Carol paid $10,000 for 5/3 loaves or $6000 per loaf. So Alice
gets $8,000 for her 4/3 loaves she gave up and Bob gets $2,000 for his
1/3 of a loaf.

This solution makes sense mathematically but not economically. Suppose
Alice and Bob were poor graduate students. Bob would be willing to eat
less bread and undercut the price of Alice's bread. To make this point
clearer suppose originally Alice had four loaves and Bob only one. Then
by the logic above, not only does Alice get all of Carol's $10,000 but
also $4,000 more from Bob for the 2/3 of a loaf he gets from Alice.

To do the correct math one would have to know the exact utility
functions of Alice, Bob and Carol, or set up an appropriate auction
when distributing the bread. But since the money is split after the
sharing has been done, Alice and Bob should just take $5,000 each and
be happy with it.

7 comments:

IMHO If the trip can be considered a "joint venture", then Alice indeed contributed 60% of the "seed funding". and so should get 60% of the shares. Marilyn's answer is good only if Carol has interacted with each camper separately, which goes contradictory to the story.

No i think it makes sense. If Carol ate all the bread herself then sure enough its 6k for Alice and 4k for Bob, but she got a a 1/3rd share. Now for that share she got she still received it in same ratios, so she still pays 6k and 4k respectively, but now from the ramaining Bob has an equal share to Alice, which means he has effectively bought (1/2 - 2/5)*(10/3) = 1/3 loaf from Alice, costing 2k.

Lance, there are two mathematical clues in the puzzle, "equally" and "math teacher". So the mathematical solution is the right one. As you said, economically there is not enough data.

So all you can enforce is the notion of "core". The mathematical solution is in the core.

Lance, the hypothetical situation you are considering where Bob gives $4000 to Alice is not in the core. So the core prevent you from this bad example, which convinced you to drop both mathematical and economical solution in favor of social (or communist?) solution.

Lance, there are two mathematical clues in the puzzle, "equally" and "math teacher". So the mathematical solution is the right one. As you said, economically there is not enough data.

So all you can enforce is the notion of "core". The mathematical solution is in the core.

Lance, the hypothetical situation you are considering where Bob gives $4000 to Alice is not in the core. So the core prevent you from this bad example, which convinced you to drop both mathematical and economical solution in favor of social (or communist?) solution.

The bread is gone. Carol should donate $6,000 to wilderness rescue efforts, and $4,000 to Oxfam. That would be a lot more positive than bickering over who owes who how much for a beautiful act of altruism during a survival emergency.